# Lecture - Linear Algebra MT22, X

> Source: https://ollybritton.com/notes/uni/prelims/mt22/linear-algebra/lectures/10/ · Updated: 2022-11-19 · Tags: uni, lecture

### Flashcards
What is $\text{Ker}(T)$ for $T : \Pi_n \to \Pi_n$ given by $T(v) = v'$?::
$$
\Pi_0
$$

What is $\text{Im}(T)$ for $T : \Pi_n \to \Pi_n$ given by $T(v) = v'$?::
$$
\Pi_{n-1}
$$

For a projection $T : V\to V$ given by $T(v) = w$ when $v = u + w$ for a vector space where $V = U \oplus W$, what is $\text{Ker}(T)$?::
$$
U
$$

For a projection $T : V\to V$ given by $T(v) = w$ when $v = u + w$ for a vector space where $V = U \oplus W$, what is $\text{Im}(T)$?::
$$
W
$$

Say you have a vector space $V$ with an ordered basis $\mathcal{E} = [v_1, \ldots, v_n]$ and another vector space $W$ with an ordered basis $\mathcal{F} = [w_1, \ldots, w_m]$, and a transformation $T : V \to W$. What is the dimension of the transformation matrix $A$ that maps between them?::
$$
m \times n
$$

Say you have a vector space $V$ with an ordered basis $\mathcal{E} = [v_1, \ldots, v_n]$ and another vector space $W$ with an ordered basis $\mathcal{F} = [w_1, \ldots, w_m]$, and a transformation $T : V \to W$. What does the transformation matrix $A$ in $A\underline{\alpha} = \underline{\beta}$ “do” for column vectors representing $v$ and $w$ as coordinates with respect to the bases?::
Translate the vector from one basis to another.

How can you still think in column vectors for any vector space?::
Consider column vectors of coordinates with respect to ordered bases.

What would be the column vector $\underline{\alpha}$ representing $1 + 2x - 3x^2$ with respect to the ordered basis $[1, x, x^2]$?::
$$
\left(\begin{matrix} 1 \\\\ 2 \\\\ -3 \end{matrix}\right)
$$

Say you have a vector space $V$ with an ordered basis $\mathcal{E} = [v_1, \ldots, v_n]$ and another vector space $W$ with an ordered basis $\mathcal{F} = [w_1, \ldots, w_m]$, and a transformation $T : V \to W$. If $A$ is the transformation matrix, then what is the $k$-th column?::
The coefficients of $T(v_k)$ with respect to $\mathcal{F}$.

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